The upper bound consists of two parts. A relatively straightforward part is given in Proposition REF  in the appendix, which shows that under stopping rule (REF ) with \(\delta =c\) , \(\mathbb {E}[\Delta _{\tau } \mid \theta = \theta _0] \leqslant O_{\theta _0}(c) = o_{\theta _0}(c\log (1/c))\) . Establishing the upper bound then requires bounding the average sample size. Past work shows us how to bound the average sample size in problems without contexts.
Arguments in [1]} allow one to show formally that \(p_{t,i} \rightarrow p^*_{i}\)  almost surely for each \(i\in [k]\) . If one makes the asymptotic approximation
\(Z_{t, I^*, j}^2 \approx t \cdot \frac{ \left(\mu (\theta , I^*, w) - \mu (\theta , j, w)\right)^2 }{ \Vert  X_{\rm pop}\Vert _{A}^2 \left( (p^*_{I^*})^{-1} + (p^*_{j})^{-1}\right)} = 2\cdot t \cdot \Gamma _{\theta }^{-1}\) 
